The Hardest Logic Puzzle Ever The Hardest Logic Puzzle Ever is a logic puzzle invented by American philosopher and logician George Boolos and published in The Harvard Review of Philosophy in 1996. A translation in Italian was published earlier in the newspaper La Repubblica, under the title L'indovinello più difficile del mondo. The puzzle is inspired by Raymond Smullyan. It is stated as follows: Three gods A, B, and C are called, in no particular order, True, False, and Random. True always speaks truly, False always speaks falsely, but whether Random speaks truly or falsely is a completely random matter. Boolos provides the following clarifications:[1] a single god may be asked more than one question, questions are permitted to depend on the answers to earlier questions, and the nature of Random's response should be thought of as depending on the flip of a coin hidden in his brain: if the coin comes down heads, he speaks truly; if tails, falsely.[2] History[edit] The solution[edit] Boolos' question was to ask A:
Relational Logic: Satisfaction and Logical Entailment We observed that a lot of the students are struggling with Problem 6.5. Therefore, we've provided the following exercise that presents four different variants of the Problem 6.5 along with hints regarding the satisfiability and entailment of the supplied relational logic sentences. Our hope is that after trying out and / or seeing the answers of this exercise, the students can better understand the relationship and the differences between consistency (also called satisfiable) and logical entailment of sentences. You can access the four variants of Problem 6.5 by choosing the appropriate option from the following dropdown menu. By default, Variant 1 is loaded. Let's suppose that our vocabulary consists of the object constants: a, b and c and the binary relation constants p and r. Consider the following sentence about r. r(a, b) ∧ r(b, c) Say whether each of the following sentences is (a) consistent with the above sentence about r, and (b) logically entailed by the above sentence about r. 1.
A Table That Turns Your Kitchen Into Mini Ecosystem [UPDATED] | Fast Company - StumbleUpon Convenience and efficiency are king when it comes to product design. What could be more efficient than a natural ecosystem? That's the insight behind a "living kitchen" designed by the brilliant young design studio Studio Gorm. They looked at what we have in our kitchens--fruits, vegetables, organic waste--and figured: That's actually enough to create a miniature system for watering fresh herbs, composting the waste, and generating new soil. None of the elements is brand new to this product, but their integration wins points for ergonomics and ease. Maybe what's most surprising is that Studio Gorm isn't based in the Netherlands or Scandinavia--but rather in Eugene, Oregon. Check out some of Studio Gorm's other designs, including a modular furniture system of pegs and boards; an elegant Egyptian-inspired chair; a handsome adjustable lamp; and an overhead light inspired by--of all things--a falafel container.
The Art of Meditation / Stop Being a Zombie! A person who thinks all the time has nothing to think about except thoughts. So he loses touch with reality, and lives in a world of illusion – Alan Watts Tweet This Have you ever driven your car or bicycle and suddenly you wake up somewhere down the road and can’t remember how you got there? What happened in those few minutes? When you put some effort in it you probably remember some of the thoughts you had. And when you are honest about it, it was probably a fantasy. And so did I when I first started meditating. So before I continue I’d like everyone to experience this to understand what I’m talking about. Did you do the two minutes? There’s a lot been written on meditation, and to be honest, I ain’t got anything new or groundbreaking stuff to tell. What is Meditation? Meditation is the act of training the mind. Why Meditate? The Buddha said: don’t blindly believe what others say, see for yourself what brings serenity, clarity of thought and inner peace. How to Meditate?
[ wu :: riddles(easy) ] So, an eccentric entrepreneur by the name of Alphonse Null has sent out a press release about his new, mind-blowing hotel: The Hotel Infinity. Null informs the world that this hotel has an infinite number of rooms (specifically, an infinity equal to the cardinality of the integers). A quick tour puts skeptics' claims to rest; as far as anyone can tell, this hotel has infinite rooms. The consequences are mind-boggling, and Null sets up a press conference to answer questions... "So, Mr. Null, how will patrons get to their room, if their room number has, say, more digits than protons in the universe?" "The elevators have an ingenious formula device instead of buttons... simply input the formula for your room number, with Ackermann numbers or somesuch... your room formula can be picked up at the front desk. "How do you produce the power and water for this hotel?" "I have infinite generators and wells, of course. "What about costs? "That's the beauty of it! "But, Mr. "Oh?"
Building Models Introduction to Logic 1. Introduction In Relational Logic, it is possible to analyze the properties of sentences in much the same way as in Propositional logic. The main problem in doing this sort of analysis for relational logic is that the number of possibilities is even larger than in propositional logic. Fortunately, as with Propositional Logic, there are some shortcuts that allow us to analyze sentences in relational logic without examining all of these possibilities. 2. In the Boolean model approach, we write out an empty table for each relation and then fill in values based on the constraints of the problem. As an example, consider the Sorority problem introduced in Chapter 1. In this particular case, it turns out that there is just one model that satisfies all of these sentences. The data we are given has three units - the fact that Dana likes Cody and the facts that Abby does not like Dana and Dana does not like Abby. Now, we know that Abby likes everyone that Bess likes. 3.
The 16-Year-Old Baby - Real People Stories As soon as Brooke Greenberg enters her summer-school classroom, she starts eyeing the door to leave. Her noisy classmates irritate her. She pays little attention to her teacher. "She loves to be on her own, away from everybody," says her mother, Melanie. Her behavior almost makes her sound like the teenager she officially is, but Brooke is like no other 16-year-old in the world. When she roams the hallways, she scoots in a pint-size walker; when she watches her favorite show, SpongeBob SquarePants, she does so from a crib. Brooke came to the world's attention eight years ago, when news organizations, including PEOPLE (Sept. 17, 2001), shared the story of this seemingly ageless child and her family in suburban Baltimore. For years Brooke's routine consisted of doctor visits and hospital stays. Brooke's story fascinated Dr. No one has felt the impact of Brooke's condition more than Melanie. No one knows how long Brooke will live. Melanie long ago stopped looking for answers.
Law of Attraction – Take Control of Your Life Warning: This article will sound like complete bullshit to the conservative mind. If you can go into the reading with an open mind and are willing to try the methods described with the intention of attracting good things to your life, please proceed. However, if you are already swaying towards skepticism, please save this article for a time when you are dying for a change or confident that this method could work for you. This might be the most important and eye-opening article you will read in your entire life so plan accordingly. Thank you Introduction to The Law Of Attraction: Many of you have probably heard of the Law of Attraction, The Secret, visualization, or some of the other popular manifestations of this technique, but most likely do not fully know what it is. Basics: Manifesting your ideal is not as hard as it sounds. 1. Other Tips: - Positive things come to positive people, so practice a good demeanor about life. 1. The New Psycho-Cybernetics by Dr. 2. 3.
Be-Fitched! Constructing proofs using the Fitch system can often be hard and unintuitive, especially for those who encounter it for the first time. We have identified the following guidelines which are based on the properties of the Goal or of the Premises that could potentially help you with Fitch-style proofs. Guidelines based on propeties of the Goal: Goal is of the form φ ⇒ ψ Assume φ Prove ψ Apply Implication Introduction to prove φ ⇒ ψ Goal is of the form ¬φ Assume φ Find a sentence ψ Prove φ ⇒ ψ Prove φ ⇒ ¬ψ Apply Negation Introduction to prove ¬φ The idea behind G2 is to assume φ and show that this leads to a contradiction i.e. ψ and ¬ψ. Good candidates for ψ when applying Negation Introduction on φ ⇒ ψ and φ ⇒ ¬ψ are the premises or the assumptions (for the sub-proof). Consider the following example. Guidelines based on propeties of Premises: Comprehensive example Apply G1-G5 to reverse-engineer which sub-goals to prove prior to proving the goal.
Mendelson Proof Tips Contradiction Realization (CR): (¬φ ⇒ ψ) ⇒ ((¬φ ⇒ ¬ψ) ⇒ φ) Observation: CR = application of Negation Introduction to prove ¬¬φ followed by Negation Elimination to prove φ. Good idea to use CR when the goal contains no implications, or if the premises have negations. (For e.g. consider Problem 4.1). For example in problem 4.2, we have to prove p from the premise ¬¬p. Note that in the Fitch system, one would start by assuming ¬p;, re-iterate ¬¬p, and finally apply Implication Introduction.
Combinational Circuit Diagnosis Once again, consider the Combinational Circuit Verification example. The focus there is on a full adder, i.e. a combinational circuit with the schematic diagram shown below. The example shows how to model this circuit in the form of logical sentences that describe the behavior of the individual gates. Of course, this description assumes that all of the components are behaving correctly. One advantage of doing things this way is that we can use logical deduction to diagnosis hardware failures. Using this data together with the sentences above and using logical deduction, we can derive the following conclusions. the first conclusion below comes from the erroneous sum bit, and the second comes from the erroneous carry bit. A common assumption in hardware diagnosis is that at most one component is failing.
Combinational Circuit Verification A combinational circuit is a collection of interconnected digital components called gates. Gates have inputs and outputs. When Boolean signals (1 or 0) are applied to the inputs of a gate, the circuit produces a corresponding output depending on the type of the gate. The following illustration is a schematic diagram for a combinational circuit called a full adder. The purpose of a full-adder is to do one slice of binary addition. We can encode the behavior of the individual gates in a circuit like this using the language of Propositional Logic. As mentioned above, the purpose of a full adder is to compute one slice of binary addition. Once we have expressed things in this way, we can use the tools of Propositional Logic to verify that the circuit works correctly.
Whodunnit Victor has been murdered, and Art, Bob, and Carl are suspects. Art says he did not do it. He says that Bob was the victim's friend but that Carl hated the victim. Bob says he was out of town the day of the murder, and besides he didn't even know the guy. Carl says he is innocent and he saw Art and Bob with the victim just before the murder. Assuming that everyone - except possibly for the murderer - is telling the truth, encode the facts of the case so that you can use the tools of Propositional Logic to convince people that Bob killed Victor. The key to using Logic to solve this problem is to make the suspects' statements conditional on their innocence. Given this vocabulary, we can encode the facts of the case as shown below. In addition, we can encode some general facts as shown below. Finally, we make the assumption that there is only one guilty party. Once we have formalized the problem in this way, we can use the tools of Propositional Logic to prove Bob's guilt.